Numerical solution of fractal-fractional differential equations system via Vieta-Fibonacci polynomials fractal-fractional integral operators

The main idea of this work is to present a numerical method based on Vieta-Fibonacci polynomials (VFPs) for finding approximate solutions of fractal-fractional (FF) pantograph differential equations and a system of differential equations. Although the presented scheme can be applied to any fractional integral, we focus on the Caputo, Atangana-Baleanu, and Caputo-Fabrizio integrals with due to their privileges. To carry out the method, first, we introduce FF integral operators in the Caputo, Atangana-Baleanu, and Caputo-Fabrizio senses. Then, by applying the Vieta-Fibonacci polynomials and their FF integral operators together with the collocation method, the problem becomes reduced to a system of algebraic equations that can be solved by Mathematical software. In the presented scheme, acceptable approximate solutions are achieved by employing only a few number of the basic functions. Moreover, the error analysis of the presented method is investigated. Finally, the accuracy of the presented method is examined through the numerical examples. The proposed scheme is implemented for some famous systems of FF differential equations, such as memristor, which is a fundamental circuit element so called universal charge-controlled mem-element, convective fluid motion in rotating cavity, and Lorenz chaotic system.